Complete Graphical Criterion for Sequential Covariate Adjustment in Causal Inference

Covariate adjustment, also known as back-door adjustment, is a fundamental tool in causal inference. Although a sound and complete graphical identification criterion, known as adjustment criterion (Shpitser et al., 2010), exists for static contexts, sequential contexts present challenges. Current practices, such as the sequential back-door adjustment (Pearl and Robins, 1995) or multi-outcome sequential back-door adjustment (Jung et al., 2020), are sound but incomplete; i.e., there are graphical scenarios where the causal effect is expressible via covariate adjustment, yet these criteria do not cover. In this paper, we exemplify this incompleteness and then present the sequential adjustment criterion , a sound and complete criterion for sequential covariate adjustment. We provide a constructive sequential adjustment criterion that identifies a set that satisfies the sequential adjustment criterion if and only if the causal effect can be expressed as a sequential covariate adjustment. Finally, we present an algorithm for identifying a minimal sequential covariate adjustment set, which optimizes efficiency by ensuring that no unnecessary vertices are included.